MUTUAL IMPEDANCE OF GROUNDED WIRES 163 



^^ ^ 27rXij„ L "^ A[aiX2 + aaXi + (aiX2 - aaXOe-^"".]] 



X Jo(ru)du, 



Pir) =2 P I [ai + a2 + (ai - a2)e--^'>"^2Jo(m)du. 



For wires lying in the plane of separation, at the depth b: 



nlai + ?^ + (ai — u)e~~'"'^'] 

 DM - J- /''' X [ai + «2 + («i - a2)g-^'"'] + 4«i^«2g-''''"^ . . s , 

 ^^''^ 2t Jo A[aiX2 + aaXi + (^1X2 - a2Xi)e-2^«>] -^olmj^w, 



Pir) =2 f" -" [ai + M + (ai - u)e-^''"qja{ru)du. 

 Jo ^ 



In these formulas: 



i = \ — 1 = imaginary unit, 



CO = Irf — radian frequency, 



A = (0:1 + 0:2) (w + Q!i) + (ai — «2)(z^ — ai)e~^^"i, 

 aj2 = w^ + i4:Tco\j (j = 1 and 2), 

 /o = Bessel function of the first kind, zero order. 



Expression (I) is identical in general form with the formula for 

 mutual impedances of grounded wires given by R. M. Foster ^ and 

 the Q{r) and P{r) functions for the two cases above reduce to agree- 

 ment with his formula, with appropriate changes in notation where 

 necessary, in any of the cases resulting in wires on the surface of 

 homogeneous earth, namely, for the first pair of functions, (i) Xi = X2, 

 (ii) b = 0, (iii) & = 00, and for the second, (i) b = 0, (ii) Xi = 0, 

 (iii) b = 00 , X2 = 0. 



It may be noted that the integrations involving the Q(r) function 

 are accomplished by inserting the four limits, which are the four 

 distances between wire terminals, since each of the indicated integra- 

 tions has a corresponding differentiation. Symbolically the result of 

 carrying out the integrations in (I) may be written as follows: 



Zss = Qu-B)(.a-b) + i(^Nss, (II) 



where 



(3(A-B)(a-6) = r r^^dSds = Q(Aa) - Q(Ab) + QiBb) - Q{Ba) 



^ R. M. Foster, "Mutual Impedances of Grounded Wires Lying on the Surface 

 of the Earth," Bulletin of the American Math. Soc, Vol. XXXVI, pages 367-368, 

 May, 1930. Bell System'Technical Journal, Vol. X, pages 408-419, July, 1931. 



